Novel Mechanism for Production of Planck Relics in Third Order Effective Field Theory

TL;DR

This work asks whether black holes can end as stable Planck-scale relics when gravity is extended by third-order effective-field-theory corrections. By deriving a corrected Schwarzschild solution and a corresponding Hawking temperature to first order in the cubic coefficient $c_6$, the authors obtain a modified evaporation law that yields two distinct halt mechanisms: for $c_6<0$, $T_H$ vanishes at $M_{crit}^- = (-2\pi c_6)^{1/4} M_p$, and for $c_6>0$, the mass-loss rate vanishes at $M_{crit}^+ = (2\pi c_6)^{1/4} M_p$ due to a truncation-induced cancellation; in both cases evaporation time diverges and Planck-scale remnants form. The analysis also reveals sign-dependent heat-capacity behavior, with a positive heat capacity below a scale for $c_6<0$ and a maximum for $c_6>0$, contrasting with Hawking’s classical result. Collectively, the results show that cubic curvature corrections alone can produce evaporation freeze-out and stable Planck-scale relics, with potential implications for the black hole information paradox and dark-matter phenomenology, while acknowledging the limitations of truncating EFT near the Planck regime.

Abstract

We present a novel mechanism for the formation of Planck-scale black hole remnants, or Planck relics. We use the third-order effective field theory corrections to the Schwarzschild geometry and Hawking temperature obtained in the literature to construct a modified evaporation law that departs significantly from Hawking's classical prediction at small masses. At a critical mass $M_{crit}$, the evaporation process comes to a halt: for $c_6 > 0$ the Hawking temperature remains finite while the leading-order mass-loss rate vanishes, whereas for $c_6 < 0$ the Hawking temperature itself drops to zero. In both cases the black hole mass approaches $M_{crit}$ only asymptotically, so that the evaporation time diverges and the objects become stable remnants. The corrected heat capacity exhibits qualitative departures from Hawking's result near the Planckian regime, with a sign change for $c_6 < 0$ and a shifted extremum for $c_6 > 0$. Our results show that cubic curvature corrections in effective field theory naturally give rise to black hole remnants without invoking ad hoc modifications to Hawking radiation.

Figures (3)

• Figure 1: Hawking temperature as a function of the normalized mass $\frac{M}{M_p}$ for three cases: the standard Hawking result ($c_6 = 0$), and third-order effective field theory with $c_6 = +1$ and $c_6 = -1$. For $c_6 < 0$, the temperature reaches a maximum at the scale $M_T^{-}$ and subsequently decreases toward zero as $M \rightarrow M_{crit}^{-}$, indicating evaporation freeze-out.
• Figure 2: Comparison of the evaporation process in third-order effective field theory for positive and negative $c_6$ as a function of the normalized mass $\frac{M}{M_p}$, shown alongside Hawking’s evaporation law.
• Figure 3: Comparison of the heat capacity $\frac{dM}{dT_H}$ as a function of the normalized mass $\frac{M}{M_p}$ of a Schwarzschild black hole in third-order effective field theory for positive and negative $c_6$, together with Hawking’s prediction. For negative $c_6$, the heat capacity becomes positive at the mass $M_T$.